In this paper we focus on Block Graph Editing, Block Graph Deletion and all three
variants of modification problems towards striclty chordal graphs. Block graphs
are a well-studied subclass of chordal graphs, that are diamond-free (Figure REF ).
They are moreover the graphs in which every biconnected component induces a clique. Notice that this observation allows for a simple polynomial-time algorithm for Block Graph Completion since one needs to turn every biconnected component of the input graph into a clique.
Strictly chordal graphs are another subclass of chordal graphs, also known as block duplicate graphs [1]}, [2]}, [3]}. They can be obtained from block graphs by repeatedly choosing some cut-vertex \(u\)  and adding a true twin \(v\)  of \(u\) , that is a vertex \(v\)  adjacent to \(u\)  and all neighbors of \(u\)  [3]}. They can also be characterized as dart, gem-free chordal graphs (see Figure REF  and next section) and are thus ptolemaic (i.e. chordal distance-hereditary [5]}).
Strictly chordal graphs are also known to be a subclass of 4-leaf power graphs [2]}, and a super-class of 3-leaf power graphs [7]}.
